Presumably a more accurate clock will make it much easier to detect small relativistic effects. This could be useful for e.g. measuring gravity waves. Or it may

Presumably a more accurate clock will make it much easier to detect small relativistic effects. This could be useful for e.g. measuring gravity waves. Or it may just make it possible to do various kinds of experiments on a smaller scale. An early direct test of relativity involved putting an atomic clock on a jet to measure time dilation due to relative motion. A more precise clock might allow the same test to be done with an automobile. In the end, the real answer to your question might well be something that we currently know nothing about. In other words: "If we already knew the answer to that question, we wouldn't need the clock."